SOUND VISUALIZATIONAND MANIPULATION

SOUND VISUALIZATIONAND MANIPULATION

Yang-Hann Kim and Jung-Woo ChoiKorea Advanced Institute of Science and Technology (KAIST), Republic of Korea

This edition first published 2013

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Library of Congress Cataloging-in-Publication Data

Kim, Yang-Hann.Sound visualization and manipulation / Yang-Hann Kim, Jung-Woo Choi.

pages cmIncludes bibliographical references and index.ISBN 978-1-118-36847-3 (cloth)

1. Sound-waves–Mathematical models. 2. Helmholtz equation. I. Choi, Jung-Woo. II. Title.QC243.K46 2013534.01′5153533–dc23

2013025393

Set in 9/11pt Times by Laserwords Private Limited, Chennai, India.

Contents

About the Author xi

Preface xiii

Acknowledgments xvii

Part I ESSENCE OF ACOUSTICS

1 Acoustic Wave Equation and Its Basic Physical Measures 31.1 Introduction 31.2 One-Dimensional Acoustic Wave Equation 3

1.2.1 Impedance 91.3 Three-Dimensional Wave Equation 101.4 Acoustic Intensity and Energy 11

1.4.1 Complex-Valued Pressure and Intensity 161.5 The Units of Sound 181.6 Analysis Methods of Linear Acoustic Wave Equation 27

1.6.1 Acoustic Wave Equation and Boundary Condition 281.6.2 Eigenfunctions and Modal Expansion Theory 311.6.3 Integral Approach Using Green’s Function 35

1.7 Solutions of the Wave Equation 391.7.1 Plane Wave 401.7.2 Spherical Wave 41

1.8 Chapter Summary 46References 46

2 Radiation, Scattering, and Diffraction 492.1 Introduction/Study Objectives 492.2 Radiation of a Breathing Sphere and a Trembling Sphere 502.3 Radiation from a Baffled Piston 582.4 Radiation from a Finite Vibrating Plate 652.5 Diffraction and Scattering 702.6 Chapter Summary 792.7 Essentials of Radiation, Scattering, and Diffraction 80

2.7.1 Radiated Sound Field from an Infinitely Baffled Circular Piston 80

vi Contents

2.7.2 Sound Field at an Arbitrary Position Radiated by an Infinitely BaffledCircular Piston 81

2.7.3 Understanding Radiation, Scattering, and Diffraction Using theKirchhoff–Helmholtz Integral Equation 82

2.7.4 Scattered Sound Field Using the Rayleigh Integral Equation 96References 97

Part II SOUND VISUALIZATION

3 Acoustic Holography 1033.1 Introduction 1033.2 The Methodology of Acoustic Source Identification 1033.3 Acoustic Holography: Measurement, Prediction, and Analysis 106

3.3.1 Introduction and Problem Definitions 1063.3.2 Prediction Process 1073.3.3 Mathematical Derivations of Three Acoustic Holography Methods

and Their Discrete Forms 1133.3.4 Measurement 1193.3.5 Analysis of Acoustic Holography 124

3.4 Summary 129References 130

4 Beamforming 1374.1 Introduction 1374.2 Problem Statement 1384.3 Model-Based Beamforming 140

4.3.1 Plane and Spherical Wave Beamforming 1404.3.2 The Array Configuration 142

4.4 Signal-Based Beamforming 1454.4.1 Construction of Correlation Matrix in Time Domain 1464.4.2 Construction of Correlation Matrix in Frequency Domain 1514.4.3 Correlation Matrix of Multiple Sound Sources 152

4.5 Correlation-Based Scan Vector Design 1604.5.1 Minimum Variance Beamformer 1604.5.2 Linear Prediction 164

4.6 Subspace-Based Approaches 1704.6.1 Basic Principles 1704.6.2 MUSIC Beamformer 1734.6.3 ESPRIT 180

4.7 Wideband Processing Technique 1824.7.1 Frequency-Domain Approach: Mapping to the Beam Space 1824.7.2 Coherent Subspace Method (CSM) 1844.7.3 Partial Field Decomposition in Beam Space 1854.7.4 Time-Domain Technique 1904.7.5 Moving-Source Localization 198

4.8 Post-Processing Techniques 2044.8.1 Deconvolution and Beamforming 2044.8.2 Nonnegativity Constraint 207

Contents vii

4.8.3 Nonnegative Least-Squares Algorithm 2094.8.4 DAMAS 210References 212

Part III SOUND MANIPULATION

5 Sound Focusing 2195.1 Introduction 2195.2 Descriptions of the Problem of Sound Focusing 221

5.2.1 Free-Field Radiation from Loudspeaker Arrays 2215.2.2 Descriptions of a Sound Field Depending on the Distance

from the Array 2215.2.3 Fresnel Approximation 2235.2.4 Farfield Description of the Rayleigh Integral

(Fraunhofer Approximation) 2255.2.5 Descriptors of Directivity 227

5.3 Summing Operator (+) 2305.3.1 Delay-and-Sum Technique 2305.3.2 Beam Shaping and Steering 2315.3.3 Wavenumber Cone and Diffraction Limit 2335.3.4 Frequency Invariant Radiation Pattern 2365.3.5 Discrete Array and Grating Lobes 237

5.4 Product Theorem (×) 2405.4.1 Convolution and Multiplication of Sound Beams 2405.4.2 On-Axis Pressure Response 243

5.5 Differential Operator and Super-Directivity (−) 2455.5.1 Endfire Differential Patterns 2455.5.2 Combination of Delay-and-Sum and Endfire Differential Patterns 2525.5.3 Broadside Differential Pattern 2525.5.4 Combination of the Delay-and-Sum and Broadside Differential Patterns 258

5.6 Optimization with Energy Ratios (÷) 2595.6.1 Problem Statement 2595.6.2 Capon’s Minimum Variance Estimator (Minimum Variance Beamformer) 2615.6.3 Acoustic Brightness and Contrast Control 2625.6.4 Further Analysis of Acoustic Brightness and Contrast Control 2735.6.5 Application Examples 276References 280

6 Sound Field Reproduction 2836.1 Introduction 2836.2 Problem Statement 284

6.2.1 Concept of Sound Field Reproduction 2846.2.2 Objective of Sound Field Reproduction 284

6.3 Reproduction of One-Dimensional Sound Field 2866.3.1 Field-Matching Approach 2866.3.2 Mode-Matching Approach 2886.3.3 Integral Approach 2896.3.4 Single-Layer Potential 295

viii Contents

6.4 Reproduction of a 3D Sound Field 2966.4.1 Problem Statement and Associated Variables 296

6.5 Field-Matching Approach 2986.5.1 Inverse Problem 2986.5.2 Regularization of an Inverse Problem 3056.5.3 Selection of the Regularization Parameter 309

6.6 Mode-Matching Approach 3116.6.1 Encoding and Decoding of Sound Field 3116.6.2 Mode-Matching with Plane Waves 3136.6.3 Mode-Matching with Spherical Harmonics 320

6.7 Surface Integral Equations 3376.7.1 Source Inside, Listener Inside (V0 ⊂ V , r ∈ V ) 3376.7.2 Source Inside, Listener Outside (V0 ⊂ V , r ∈ �) 3406.7.3 Source Outside, Listener Outside (V0 ⊂ �, r ∈ �) 3416.7.4 Source Outside, Listener Inside (V0 ⊂ �, r ∈ V ) 3426.7.5 Listener on the Control Surface 3426.7.6 Summary of Integral Equations 3446.7.7 Nonradiating Sound Field and Nonuniqueness Problem 344

6.8 Single-layer Formula 3466.8.1 Single-layer Formula for Exterior Virtual Source 3466.8.2 Integral Formulas for Interior Virtual Source 355References 369

Appendix A Useful Formulas 371A.1 Fourier Transform 371

A.1.1 Fourier Transform Table 371A.2 Dirac Delta Function 374A.3 Derivative of Matrices 374

A.3.1 Derivative of Real-Valued Matrix 374A.3.2 Derivative of Complex-Valued Function 375A.3.3 Derivative of Complex Matrix 376

A.4 Inverse Problem 376A.4.1 Overdetermined Linear Equations and Least Squares (LS) Solution 377A.4.2 Underdetermined Linear Equations and Minimum-Norm Problem 378A.4.3 Method of Lagrange Multiplier 379A.4.4 Regularized Least Squares 380A.4.5 Singular Value Decomposition 380A.4.6 Total Least Squares (TLS) 382

Appendix B Description of Sound Field 385B.1 Three-Dimensional Acoustic Wave Equation 385

B.1.1 Conservation of Mass 385B.1.2 Conservation of Momentum 385B.1.3 Equation of State 388B.1.4 Velocity Potential Function 390B.1.5 Complex Intensity 391B.1.6 Singular Sources 392

Contents ix

B.2 Wavenumber Domain Representation of the Rayleigh Integral 398B.2.1 Fourier Transform of Free-Field Green’s Function (Weyl’s Identity) 398B.2.2 High Frequency Approximation (Stationary Phase Approximation) 399

B.3 Separation of Variables in Spherical Coordinates 400B.3.1 Angle Functions: Associated Legendre Functions 400B.3.2 Angle Functions: Spherical Harmonics 402B.3.3 Radial Functions 404B.3.4 Radial Functions: Spherical Bessel and Hankel Functions 404B.3.5 Description of Sound Fields by Spherical Basis Function 408B.3.6 Representation of the Green’s Function 409References 411

Index 413

About the Author

Yang-Hann KimThe research area of Yang-Hann Kim is mainly acoustics, noise/vibration. Experimental approachesand associated digital signal processing are used the most. Research projects include sound fieldvisualization, noise source identification using array microphones, detection and estimation of mov-ing noise source, structural acoustics, duct acoustics, silencer design, diagnostics of machines,and active noise/vibration control. Recently, he has been recognized as a pioneer in the fieldof sound visualization and manipulation. The latter is to make any sound field or shape in theselected region/regions. Therefore, it can be used for having very focused sound field, privatesound zone/zones, or 3D listening field.

Dr Kim joined the Department of Mechanical Engineering as an Associate Professor in 1989.Previously he worked for five years at the Korea Institute of Technology as an Assistant andAssociate Professor of the Department of Mechatronics. From 1979 to 1984, he was a researchassistant at the Acoustics and Vibration Laboratory of Massachusetts Institute of Technology whilepursuing Ph.D. degree in the field of acoustics and vibration, and obtained Ph.D. in February 1985at M.I.T., Mechanical Engineering (O.E. Program).

He has been on the editorial board of Mechanical Systems and Signal Processing (MSSP), edito-rial advisor of the Journal of Sound and Vibration (JSV) and Journal of Noise Control Engineering.He also served KSNVE as an editor for three years (1995–97). His research has been recognizedin the professional societies and institutes in many respects, including the best paper award byKSNVE (1998), the best research award by ASK (1997), second place award in the sound visual-ization competition by the Acoustical Society of America (1997), the best international cooperationaward from KAIST, and KSNVE, and the best teaching award from KAIST, department of M.E(2010). He is elected as co-chairman of inter-noise 2015, San Francisco, also a director of I-INCE.He is a Fellow of the Acoustical Society of America.

Dr Kim has published more than 100 papers, mostly in the field of sound visualization andmanipulation, in the well-known journals, including the Journal of the Acoustical Society of America,Journal of Sound and Vibration, and Journal of Acoustics and Vibration, the Transaction of ASME.He is an author of well-known acoustics text, Sound Propagation: An Impedance Based Approach,published by John Wiley & Sons, Inc. He also wrote the chapter “Acoustic holography” in theHandbook of Acoustics, published by Springer Verlag. He has delivered two plenary lectures inICA (2004), and Inter-Noise (2012) and one keynote lecture in ICSV (2009). All of these lectureswere on acoustic holography, sound visualization, and manipulation.

Jung-Woo ChoiJung-Woo Choi’s primary research area includes active sound control and array signal processingfor loudspeaker/microphone arrays. His research interests also include sound field reproduction,sound focusing, and their application for audio systems. From 1999, he has been working on

xii About the Author

sound/noise control over elected regions based on the concept of acoustic contrast, which hasbeen widely adopted for the implementation of personal sound zones. Recently, his research hasextended to interactive 3D sound/multi-channel audio systems that can be manipulated in real-timeby exploiting the beauty of direct integral formulas.

Dr Choi received his B.Sc., M.Sc., and Ph.D. degrees in Mechanical Engineering from the KoreaInstitute of Science and Technology (KAIST), Korea, in 1999, 2001, and 2005, respectively. Hewas a Postdoctoral Research Associate with the Center for Noise and Vibration Control (NOVIC),KAIST, Korea, 2005–06. From 2006 to 2007, he was a Visiting Postdoctoral Researcher at theInstitute of Sound and Vibration Research (ISVR), University of Southampton, UK. From 2007 to2011, he was with Samsung Electronics at the Samsung Advanced Institute of Technology (SAIT)in Korea, working on array-based audio systems as a Research & Development staff member anda Senior Engineer. In 2011, he joined the Department of Mechanical Engineering, KAIST, Korea,and has since been a Research Professor there. He is the author of more than 50 papers/conferencearticles and 15 patent applications, including five registered patents on loudspeaker array systems.

Preface

If only we could see sound propagation in space with our eyes, and if only the sound couldbe created in any desired shape! Such a fantastic concept is being realized. New approaches toacoustics and noise engineering have allowed innovative changes in these fields.

So far, extensive efforts have been made using various methods to explain how a medium changesas sound propagates in space or how the shape of the sound propagation changes depending onits frequency and wavelength. There are two main approaches being employed to resolve thesequestions: theoretical and experimental. The theoretical approach is adopted to develop an under-standing of the phenomena of sound propagation in acoustic waves and, through this understanding,to attempt to find a solution. The characteristics of acoustic wave equations in certain cases areinterpreted by numerically solving the so-called linear acoustic wave equations. Popular numericaltechniques are the finite element method and boundary element method; both have achieved incred-ible developments owing to continuous evolutions in their background theories and improvementsin the arithmetic capacity of computers. The experimental approach has also seen rapid improve-ments. Developments in semiconductor technologies have reduced the microphone size to eliminateunnecessary scattering induced by them, and the reduction in cost allows for tens and hundreds ofmicrophones to be used at the same time. We can now sample, record, and analyze signals fromhundreds of microphones in almost real time. These developments allow us to “visualize” soundusing our eyes in the real world, which is something that human beings have long dreamt of. Thefirst half of this book explains various methods to visualize sound.

From a mathematical point of view, sound visualization can be regarded as an exploration ofmethods to transform measured data into information that is visible to the human eyes. Mostinformation transformation is determined during selection of the desired basis function becauseinformation transformation can produce different results depending on its mapping functions, asshown in the figure below. Thus, we need to deal with problems such as selecting a basis functionand expressing a sound field as a visible image using the selected basis function. This book explainsthe planar, cylindrical, and spherical basis functions used in acoustic holography and the func-tions employed for beamforming methods. Their advantages and disadvantages as well as practicalapplicability are addressed. The advantage of the acoustic holography method includes visualiza-tion of information with great physical significance, such as acoustic pressure, velocity, intensity,and energy. On the other hand, the beamforming method can provide a variety of visualizationinformation depending on the type of basis function used for the beamformer.

The concept that visualization results vary significantly depending on the basis functions canbe reasonably expanded to an idea of sound manipulation where arbitrary or desired forms ofsound can be created in space. Desired sounds can be produced by selecting basis functions sothat the sounds generated from the sound sources arranged in the space are of these types of basisfunctions, as shown in the figure. Well-known methods include wave field synthesis (WFS) andAmbisonics. WFS is a representative method based on the so-called Kirchhoff–Helmholtz integral

xiv Preface

: what is available(measured data)

equation, whereas Ambisonics is a technique that expresses sound fields using spherical harmonicsand embodies the desired shapes of sound in space using such expressions. From a unifying point ofview, the manipulation of a sound field is an issue in obtaining the desired output using the availablesound sources; accordingly, we can select the best basis function depending on the definition ofthe desired function. Based on this idea, the sound focusing problem of concentrating the sound indesired areas or dividing an area into acoustically bright and dark zones and maximizing the ratioof sound energies between the two areas can also be explained.

Both the sound visualization method and sound manipulation method demand considerable the-oretical knowledge of mathematics and acoustics as well as knowledge of signal processing tounderstand their principles and realize their practical applications. To aid potential readers whowant to understand the basic concepts or those who will practically apply the methods, the sim-plest one-dimensional theories are introduced in this book, and their mathematical and theoreticalexplanations are presented in every chapter. Chapter 1 is intended to aid understanding of the basicphysical quantities in acoustics. Part I consists of two chapters. Chapter 1 explains three physicalquantities in acoustics using one-dimensional examples: interrelationships among acoustic pressure,particle velocity, and acoustic density. This approach is justified in that the principle of superpo-sition holds for a linear system; hence, most of the concepts explained in one dimension can beextended to multidimensional cases.

Part II introduces the sound visualization methods and explains how their basic principles can bevaried depending on certain basis functions. Accordingly, basis functions and approaches for theacoustic holography and the beamforming method are introduced. An appropriate basis functionshould be used depending on what we want to visualize; depending on this basis function, theinformation of visualized sound fields can be varied.

In Part III, we deal with sound manipulation techniques. Sound manipulation is carried out usingtwo main methods; both are discussed with respect to how they are embodied in one-dimensionalsituations. Sound manipulation involves a sound focusing technique that concentrates the soundin specific areas in space and a sound field reproduction method that generates a wave front inthe desired forms. For realizing these two methods, unique inputs to generate sound fields in thedesired forms need to be determined. Therefore, the sound focusing and reproduction problems aredefined as inverse problems corresponding to the beamforming and acoustic holography methods,respectively. Thus, the sections on acoustic holography and sound field reproduction are organizedto complement each other. The sections on beamforming and sound focusing address similar issuesbut explain them from different points of view. The chapter on beamforming focuses on a signalprocessing technique to extract the parameter determining the locations of sound sources, whereasthe chapter on sound focusing explains resolution variations depending on the geometric configura-tion of arrays and beam pattern variations depending on the basic aperture functions. Thus, Parts II

Preface xv

and III address different and similar issues from complementary points of view; readers interestedin visualization are strongly recommended to read the manipulation part. It would be efficient forreaders of this book to use Part I as a reference when they need to know more about acousticswhile reading Parts II and III.

In conclusion, this book introduces and explains methods for sound visualization and manipula-tion. The book is organized such that readers can gain a profound understanding of basic conceptsand theoretical approaches from the one-dimensional case. The methods of visualization and manip-ulation are explained as a unifying approach for creating certain assumed or desired shapes in spacebased on the measured or available information using basis functions.

Yang-Hann KimJung-Woo Choi

Acknowledgments

It was around 1990 that the first author had an idea about the basis function described in this book.He visited his old friend, Prof. J. K. Hammond of the Institute of Sound and Vibration Research(ISVR), University of Southampton, who was giving a lecture on nonlinear signal processing fora group of people in the industry at the time. The first page of the handout made for that classincluded a primitive version of the picture that is published in this book’s preface. In fact, thispicture originated and evolved from the image in Science with a Smile (Robert L. Weber, Instituteof Physical Publishing, Bristol and Philadelphia, 1992, pp. 111–12). The moment he looked at thispicture, it occurred to him that a part of this picture could be used to explain the processing of everysignal. Signal processing essentially involves finding desired information using available data. Thus,this picture symbolically shows that the quality of information obtained eventually depends on howwell the processing method represents substantial, physical, or mathematical situations. Ultimately,the result is fully dependent on the processing method one has chosen, that is, a basis function. Ifso, how do we select a basis function? Although it is a very basic question, it is self-evident thatif we can suggest the best method to be selected, it would serve as a very innovative and usefulmethod in this discipline.

In fact, since the first author was at the time working on issues such as mechanical noisediagnosis and fault detection using signal processing, he looked at the picture that Prof. Ham-mond had used in a symbolic manner and gained an idea to view various problems in a unifyingmanner. He came to realize that both sound visualization techniques – acoustic holography andbeamforming – eventually produced different results with regard to visualization owing to differ-ences in the basic functions that were used. If so, in-depth knowledge of whether a basis functionused has a mathematical function to perform something well would allow one to have a good under-standing of the result of the sound visualization, that is, the picture. Thus, to clearly interpret thevisualized information gained through the acoustic holography method and to accurately analyzethe desired information, it is necessary to analyze how well the basis function expresses the desiredvisual information. Similarly, the following questions can be approached from an understandingof the basis function that was used: what is the specific information that can be gained from thespatial distribution of the beamforming power obtained from the beamforming method? Does themaximum value of the beaming power correctly describe the locations of sound or noise sources?What properties of the sources does the spatial distribution of the beaming power represent? Inthis regard, the visualization described in this book was greatly inspired by the discussions withProf. Hammond.

The sound manipulation study described in the second half of this book started, it is recalled,around 1999 when the first author thought he had found some improvements in the study of visu-alization or was getting bored with studying sound visualization. The second author was studyinghow to focus sounds in an arbitrary space as part of the work for a master’s degree, and basedon this, he started a full-scale study on sound manipulation. The first result aimed to practically

xviii Acknowledgments

implement a system that allows one to hear a desired sound without disturbing others by focusingthe sound in a specific space. Fortunately, the experiment was successful, and in 2000, the authorssucceeded in focusing sound in a specific space by using six loudspeakers. Later on, this study wasexpanded to a study of a home/mobile speaker array system by the second author in the industryand to another study of monitor speaker array development by the first author at KAIST. Themonitor speaker array system led to the implementation of a personal audio system that focusessound using nine speakers. Among those who participated in the theoretical development and theexperiment are Chan-hee Lee, currently working at Hyundai Heavy Industries Co., Ltd.; Dr Ji-hoChang, currently at DTU after completing his doctoral degree; and Jin-young Park, currently pur-suing his doctoral degree. At that time, the result attracted so much attention that it was broadcaston national TV. Thanks to this, the first author was granted an unexpected research fund, and histeam could build a set of experimental equipment consisting of 32 speakers supported by KAIST’sHRHR project. Using this experimental equipment, the research team implemented a method forfocusing sounds on specific spots in various ways, and the effects were found to be better thanexpected. One day, a question was raised about what results would be produced if the focusingpoint was moved to an arbitrary location. Min-ho Song developed an interface using an iPhone, inwhich as a finger moved to a location, the location at which sound was focused was also changed,making the sound audible in real time. In fact, from a theoretical viewpoint, they knew that thesound focusing solution had nothing to do with 3D sound. However, the listener could feel theeffect of the location of the sound source moving through the sound focusing solution only. As amatter of fact, studies have reported on a focused source using the time-reversed driving functionin wave field synthesis, but these did not have sufficient theoretical basis, and no perfect integralequation form was available for the array in the form of surrounding the listener. The theoreticalexplanation of the experimental results was completed by the second author in 2011, and it wasproved that a general solution can be drawn by combining Porter-Bojarski integral with a multipolevirtual source. The first work to create sounds using this solution aimed to relocate a mosquito’ssound to a desired space, and it was a great success. This substantial success became a motivationfor the book’s third part. The doctoral students, Jeong-min Lee and Dong-su Kang, made substantialcontributions to developing a speaker system that implemented a sound ball. In addition, the authorswould like to acknowledge Dr Min-Ho Song, who performed great research while completing hisdoctoral degree at the Graduate School of Cultural Technology and who contributed to developingone particular interface.

In fact, the graduates’ wonderful studies were greatly helpful to the authors in writing the soundvisualization part. In particular, studies by Dr Jae-Woong Choi, who has made great achievementsin the areas of spherical beamforming and music, and Dr Young-Key Kim, who founded a companyand has been disseminating sound visualization technology, were very helpful in writing the beam-forming chapter. Dr Hyu-sang Kwon developed moving frame acoustic holography (MFAH), and heis expected to realize great achievements as an expert in this area. Furthermore, Dr Soon-hong Parkof the Korea Aerospace Research Institute has made a great contribution to the method by applyingMFAH to moving sound sources. The authors also want to acknowledge Dr Sea-Moon Kim of theKorea Institute of Ocean Science and Technology who successfully lead the acoustic holographyexperiment on the King Seong-deok Bell; Dr Kyung-Uk Nam of Hyundai Motor Company whogreatly contributed to the partial field acoustic holography; and Dr Chun-Su Park who developedthe time domain acoustic holography technique using the spatio-temporal complex envelope.

Credit for Chapter 4 of this book also belongs to Ku-Hwan Kim, who programmed most of thecodes for beamforming simulations. The authors would like to express their appreciation to all thelaboratory members – Jung-Min Lee, Dong-Soo Kang, Dae-Hoon Seo, Ki-Won Kim, Myung-RyunLee, Seong-Woo Jung – for their enthusiasm in correcting errors and giving advice to improve thecontent of this book.

Acknowledgments xix

The second author also wants to thank his former advisors at KAIST – Yoon-Sik Park,Chong-Won Lee, Jeong-Guon Lee, and Young-Jin Park – for teaching him the fundamentals ofsound and vibration. Special thanks must be given to Prof. P. A. Nelson and S. J. Elliott andDr F. M. Fazi of ISVR for many hours of fruitful discussions with him regarding the sound fieldreproduction and sound focusing projects. The experiences with his former colleagues at SamsungElectronics – Youngtae Kim, Jungho Kim, Sang-Chul Ko, and Seoung-Hun Kim – were greatlyhelpful in summarizing the techniques discussed in Chapters 4–6.

Finally, the authors would like to express their special thanks to James Murphy and Clarissa Limof John Wiley & Sons for their consistent help and cooperation with regard to editing this book.Without their encouragement, this book would not have been possible.

Yang-Hann KimJung-Woo Choi

Part IEssence of AcousticsSound is an important part of our lives. Even in the womb, human beings are capable of detectingsounds. We create and enjoy sounds, and we can identify information conveyed by sound. Welive with sound and are familiar with the fundamental concepts associated with it. Fundamentalconcepts of sound visualization and manipulation can also be explained on the basis of themechanisms by which a sound wave is generated, propagated, and decayed by various internaland external disturbances.

Acoustics is a vast field of study that explains the propagation of waves in different media, and itcannot be completely covered in merely the first two chapters of this book. In this book, however,we limit the scope by focusing on the general idea of acoustics in terms of its essential physicalmeasures. This part of the book discusses essential measures that refer to the primary measures orphysical variables used in acoustics, such as acoustic pressure, velocity, intensity, and energy, whichcan be used to describe sound propagation. Various impedances, radiations, scatterings, surfaces,and so on, are also considered as important measures that affect wave propagation in space. In orderto uniquely and conveniently explain the physics of acoustics, this part of the book relies heavilyon the concept of impedance as a window to study sound propagation in time and space.

Chapter 1 introduces the essential physical parameters used in acoustics measurements. Thesignificance of physical parameters other than impedance, such as sound pressure, speed, energy,power, and intensity, are explained (Figure I.1). It is emphasized that these parameters form thefundamental concepts required for understanding the propagation of sound waves. The aforemen-tioned parameters are explained by using a one-dimensional approach. The Euler equation is usedto describe the relation between the sound pressure and the particle velocity in a given medium.The state equation, on the other hand, is used to evaluate the relation between the acoustic densityand the fluctuating pressure, which is the acoustic pressure that causes the sound propagation. Thethird equation used is the law of conservation of mass for the compressible fluid, which defines howthe fluctuating density and the fluid particle velocity are associated with each other. Therefore, thethree essential variables: sound pressure, particle velocity, and fluctuating density, follow these threeequations. This enables the derivation of the acoustic wave equation that governs all the parametersassociated with acoustic wave propagation. Chapter 1 discusses the two different approaches thatcan be used to solve this acoustic wave equation. The first one is based on the eigenfunction analy-sis, in which the solution is determined as the superposition of eigenmodes. Another approach usesGreen’s function, which describes how a sound field is constructed when the field has a monopolesource at an arbitrary position in space. This approach leads to the Kirchhoff–Helmholtz equation.

Sound Visualization and Manipulation, First Edition. Yang-Hann Kim and Jung-Woo Choi.© 2013 John Wiley & Sons Singapore Pte. Ltd. Published by John Wiley & Sons Singapore Pte. Ltd.

2 Essence of Acoustics

velocityparticle velocity

Equation of state Linear Euler's equationacoustic pressure (p′)

density (r0, r′) particle velocity (u)

mass conservation

=∂r′∂ t

∂u–r0∂x

= c2p′r′

− =∂p′∂x

∂ur0 ∂ t

r = r0 + r′density

p = p0 + p′acoustic pressureaccess pressuresound pressure

Figure I.1 Pictorial relation between three variables that govern acoustic wave propagation (p0 and ρ0 expressthe mean pressure and static density, respectively; p′ and ρ ′ denote acoustic pressure and fluctuating density,respectively; c denotes the speed of propagation, and u is the velocity of the fluctuating medium)

= +

incidentwave

reflectedwave

transmittedwave

radiationpressure

blockedpressure

rigid wall

Figure I.2 Reflection and transmission phenomena using the principle of superposition

Chapter 2 takes a rather ambitious route to describe how sound wave reacts under impedancemismatch in space and time using the concept of radiation, scattering, and diffraction. It is believedthat the scattering and the diffraction of sound can be explained by acoustic radiation. For instance,a scattered sound field is a result of radiation scattering (Figure I.2), whereas diffraction is a resultof the radiation from an object that has spatial impedance mismatch.

An understanding of the first two chapters is expected to help in analyzing and explaining theresults obtained by the sound visualization described in Chapters 3 and 4, and the manipulationdescribed in Chapters 5 and 6.

1Acoustic Wave Equation and ItsBasic Physical Measures1

1.1 IntroductionThe waves along a string propagate along its length, but the string itself moves perpendicular to thepropagation direction. It therefore forms a transverse wave. If the particle of a medium moves inthe direction of propagation, we refer to it as a longitudinal wave. The waves in air, water, or anycompressible medium are longitudinal waves, which are often referred to as acoustic waves. Thischapter explores the underlying physics and sensible physical measures related to acoustic waves,including pressure, velocity, intensity, and energy. Impedance plays a central role with regard toits effect on these measures.

In the area of sound visualization, our objectives are to determine a rational means to convertessential acoustic variables such as pressure, velocity, and density, or other physically sensibleacoustic measures such as intensity or energy, into visible representations. One very straightforwardway to accomplish these objectives is to express acoustic pressure by using a color code. Notably,there are many ways to visualize a sound field, depending on a mapping or general basis function,which relates acoustic variables to visual expressions. Therefore, this chapter starts with a discussionon the visualization of a one-dimensional acoustic wave.

1.2 One-Dimensional Acoustic Wave EquationThe simplest case is illustrated in Figure 1.1. The end of a pipe or duct which is filled with ahomogeneous compressible fluid (air, water, etc.) is excited with a radian frequency (ω = 2π f ,f : frequency in Hz). If the pipe is semi-infinitely long, then the pressure in the pipe (p(x, t)) canbe mathematically written as

p(x, t) = P0 cos(kx − ωt + φ) (1.1)

where P0 is the pressure magnitude and φ is an initial phase. Here, k represents the spatial frequency(k = 2π/λ, λ: wavelength in m) of the pressure field, which is often called wavenumber.

If the pipe is of finite length L, then the possible acoustic pressure in the pipe can be written as

p(x, t) = P0 cos k(L − x) cos ωt . (1.2)

1 Sections of Chapter 1 have been re-used with permission from Ref. [1].

Sound Visualization and Manipulation, First Edition. Yang-Hann Kim and Jung-Woo Choi.© 2013 John Wiley & Sons Singapore Pte. Ltd. Published by John Wiley & Sons Singapore Pte. Ltd.

4 Acoustic Wave Equation and Its Basic Physical Measures

cos ωt

pS Sdxp∂p∂x

+

x

Δ x

x xx Δ+

∂u ∂u∂t ∂x

+ uρ

This depicts the waves that can be generatedwhen we excite one end of the pipe, harmonically.

S : cross section area (m2)

ρ : density of fluid (kg/m3)u : fluid particle velocity in x − direction (m/s)

0

0

Figure 1.1 Relation between forces and motion of an infinitesimal fluid element in a pipe (expressing momen-tum balance: the left-hand side shows the forces and the right exhibits the change of momentum)

Equations (1.1) and (1.2) are different simply because of the boundary conditions: the formerhas no boundary condition prescribed at x = L, but the latter has a rigid-wall condition (velocityis zero).

To understand what is happening in the pipe, we have to understand how pressures and velocitiesof the fluid particles behave and are associated with each other. This motivates us to look at aninfinitesimal element of the volume of the fluid in the pipe; specifically, we will investigate therelation between force and motion.

As illustrated in Figure 1.1, the forces acting on the fluid between x and x + �x and its motionwill follow the conservation of momentum principle. That is,

Sum of the forces acting on the fluid = momentum change (1.3)

We can mathematically express this equality as

(pS )x − (pS )x+�x = ρSdu

dt�x (1.4)

where it has already been assumed that the viscous force, which likely exists in the fluid, is smallenough (relative to the force induced by pressure) to be neglected.

The rate of change of velocity (du/dt) can be expressed by

du

dt= ∂u

∂t+ ∂u

∂x

∂x

∂t(1.5)

where u is a function of position (x) and time (t) and velocity is the time rate change of thedisplacement. Therefore, we can rewrite Equation (1.5) as

du

dt= ∂u

∂t+ u

∂u

∂x. (1.6)

One-Dimensional Acoustic Wave Equation 5

If the cross-section between x and x + �x is maintained constant and �x becomes small(�x → 0), then Equation (1.4) can be expressed as2

−∂p

∂x= ρ

(∂u

∂t+ u

∂u

∂x

)= ρ

Du

Dt(1.7)

where

p = p0 + p′ (1.8)

ρ = ρ0 + ρ ′ (1.9)

D

Dt= ∂

∂t+ u

∂

∂x. (1.10)

Note that the pressure (p) is composed of the static pressure (p0) and the acoustic pressure (p′),which is induced by the small fluctuation of fluid particles. The density also has two components:the static density (ρ0) and the small fluctuating density (ρ ′).

Equation (1.10) is the total derivative, and is often called the material derivative. The first termexpresses the rate of change with respect to time, and the second term can be obtained by examiningthe change with respect to space as we move with the velocity u.3 As can be anticipated, the secondterm is generally smaller than the first.

If the static pressure (p0) and density (ρ0) do not vary significantly in space and time, thenEquation (1.7) becomes

−∂p′

∂x= ρ0

∂u

∂t(1.11)

where p′ is acoustic pressure and is directly related to acoustic wave propagation. As already impliedin Equation (1.8), acoustic pressure is considerably smaller than static pressure.4 Equation (1.11)essentially means that a small pressure change across a small distance (∂x) causes the fluid ofmass/unit volume ρ0 to move with the acceleration of ∂u/∂t . This equation is generally referredto as a linearized Euler equation. Equation (1.7), on the other hand, is an Euler equation.

Equations (1.7) and (1.11) describe three physical parameters, pressure, fluid density, and fluidparticle velocity. In other words, they express the relations between these three basic variables. Inorder to completely characterize the relations, two more equations are needed.

The relation between density and fluid particle velocity can be obtained by using the conservationof mass. Figure 1.2 shows how much fluid enters the cross-section at x and how much exits throughthe surface at x + �x . If we apply the principle of conservation of mass law to the fluid volumebetween x and x + �x , the following equality can be written.

⟨ the rate of mass increase in the infinitesimal element

= the decrease of mass resulting from the fluid that is entering and exiting through

the surface at x and x + �x

⟩

Expressing this equality mathematically leads to

∂

∂t(ρS�x) = (ρuS )x − (ρuS )x+�x (1.12)

2 Note that we used ∂x/∂t = u in Equation (1.6). This is the Lagrangian description, which describes the motionof a mass of fluid at �x . The other method to describe the momentum change through fixed infinitesimal controlvolume is by using the Euler description (Section 8.1.2). Note also that a more precise momentum balance can beexpressed as − ∂p

∂x= Dρu

Dt .3 We assume that the effect of mass transport is negligible.4 We also refer to the acoustic pressure as “access pressure” or “sound pressure.”

6 Acoustic Wave Equation and Its Basic Physical Measures

x x + Δx

x0

∂∂t

S : cross section area (m2)

r : density of fluid (kg/m3)u : fluid particle velocity in x − direction (m/s)

(ruS )x (ruS )x + Δx(rSΔx)

Figure 1.2 Conservation of mass in an infinitesimal element of fluid (increasing mass of the infinitesimalvolume results from a net decrease of the mass through the surfaces of the volume)

as illustrated in Figure 1.2. As assumed before, if the area of the cross-section (S) remains constant,then Equation (1.12) can be rewritten as

∂ρ

∂t= − ∂

∂x(ρu). (1.13)

We can linearize this equation by substituting Equation (1.9) into Equation (1.13). Equation (1.13)then becomes

∂ρ ′

∂t= −ρ0

∂u

∂x. (1.14)

Equations (1.11) and (1.14) express the relation between sound pressure and fluid particle veloc-ity, as well as the relation with fluctuating density and fluid particle velocity, respectively. One moreequation is therefore needed to completely describe the relations of the three acoustic variables:acoustic pressure, fluctuating density, and fluid particle velocity. The other equation must describehow acoustic pressure is related to fluctuating density. Recall that a pressure change will inducea change in density as well as other thermodynamic variables, such as entropy. This leads us topostulate that acoustic pressure is a function of density and entropy, that is,

p = p(ρ, s) (1.15)

where s denotes entropy. We can then write the change of pressure, or fluctuating pressure, dp orp′, by modifying Equation (1.15) as follows:

dp = ∂p

∂ρ

∣∣∣∣s

dρ + ∂p

∂s

∣∣∣∣ρ

ds . (1.16)

This equation simply states that a pressure change causes a density change (dρ) and entropyvariation (ds). It is noticeable that the fluid obeys the law of isentropic processes when it oscillateswithin the range of the audible frequency: 20 Hz to 20 kHz.5 The second term on the right-hand sideof Equation (1.16) is therefore negligible. This implies that the small change of sound pressure withregard to the infinitesimal change of density can be assumed to have certain proportionality. (Analternative way to deduce the same relation can be found in Appendix B, Section B.1.3.) Note thatthe second relation of Equation (1.16) is mostly found experimentally. This reduces Equation (1.16)to the form

p′

ρ ′ = B

ρ0= c2 (1.17)

5 This is possible if the period of oscillation by the fluid particle is much smaller than the time required to dissipateor transfer the heat energy within the wavelength of interest.

One-Dimensional Acoustic Wave Equation 7

where B is the bulk modulus that expresses the pressure required for a unit volume change and c

is the speed of sound. We may obtain Equation (1.17) by introducing a gas dynamics model. Thisequation is an equation of state. Tables 1.1 and 1.2 summarizes the speed of sound in accordancewith the state of gas [2]. An alternative method of deducing Equations (1.16) and (1.17) can befound in Appendix B, Section B.1.3.

Table 1.1 The dependency of the speed of sound on temperature

Temperature(◦C)

Speed of sound(m/s)

Temperature(◦C)

Speed of sound(m/s)

Temperature(◦C)

Speed of sound(m/s)

−100 263.5 −35 309.5 30 349.1−95 267.3 −30 312.7 35 352.0−90 271.1 −25 315.9 40 354.8−85 274.8 −20 319.1 45 357.6−80 278.5 −15 322.3 50 360.4−75 282.1 −10 325.3 55 363.2−70 285.7 −5 328.4 60 365.9−65 289.2 0 331.5 65 368.6−60 292.7 5 334.5 70 371.3−55 296.1 10 337.5 75 374.0−50 299.5 15 340.4 80 376.7−45 302.9 20 343.4 – –−40 306.2 25 346.3 – –

Table 1.2 The dependency of the speed of sound on relative humidity and on frequency

Relativehumidity/frequency

0% 30% 60% 100%

Decayrate(%)

Speed ofsound(m/s)

Decayrate(%)

Speed ofsound(m/s)

Decayrate(%)

Speed ofsound(m/s)

Decayrate(%)

Speed ofsound(m/s)

20 0.51 343.477 0.03 343.807 0.02 344.182 0.01 344.68540 1.07 343.514 0.11 343.808 0.06 344.183 0.04 344.68550 1.26 343.525 0.17 343.810 0.09 344.183 0.06 344.68563 1.43 343.536 0.25 343.810 0.15 344.184 0.09 344.685100 1.67 343.550 0.50 343.814 0.34 344.185 0.22 344.686200 1.84 343.559 1.01 343.821 0.99 344.190 0.77 344.689400 1.96 343.561 1.59 343.826 1.94 344.197 2.02 344.695630 2.11 343.562 2.24 343.827 2.57 344.200 3.05 344.699800 2.27 343.562 2.85 343.828 2.94 344.201 3.57 344.7011 250 2.82 343.562 5.09 343.828 4.01 344.202 4.59 344.7042 000 4.14 343.562 10.93 343.829 6.55 344.203 6.29 344.7054 000 8.84 343.564 38.89 343.831 18.73 344.204 13.58 344.7066 300 14.89 343.565 90.61 343.836 42.51 344.204 27.72 344.70610 000 26.28 343.566 204.98 343.846 101.84 344.206 63.49 344.70612 500 35.81 343.566 294.08 343.854 155.67 344.208 96.63 344.70718 000 52.15 343.567 422.51 343.865 247.78 344.211 154.90 344.70820 000 75.37 343.567 563.66 343.877 373.78 344.215 237.93 344.709

Adapted from CRC Handbook of Chemistry and Physics, 79th ed., 1998, pp. 14–38, CRC Press.: With kindpermission of Taylor & Francis Group LLC-Books

8 Acoustic Wave Equation and Its Basic Physical Measures

Note that Equation (1.17) expresses how the access pressure or acoustic pressure communicateswith the fluctuating density. Equations (1.11) and (1.14) completely express the laws that governthe waves in which we are interested. Therefore, we can summarize the relations as

−∂p′

∂x= ρ0

∂u

∂t(1.11)

∂ρ ′

∂t= −ρ0

∂u

∂x(1.14)

p′

ρ ′ = c2. (1.17)

Figure 1.3 demonstrates how these equations and physical variables are related. If we eliminateρ ′ and u from Equations (1.11), (1.14), and (1.17), then we obtain

∂2p′

∂x2= 1

c2

∂2p′

∂t2(1.18)

This is a linearized acoustic wave equation.6

Equation (1.18) is essentially a general one-dimensional acoustic wave, that is, the waves incompressible fluid. A similar relation can be found from the propagation of a string wave. Theonly difference between the waves along a string and acoustic waves lies in whether the directionsof wave propagation and velocity fluctuation of medium are collinear or perpendicular. Note thatthe propagation direction of the waves along a string is perpendicular to that of the motion of thestring. Conversely, the acoustic wave propagates in the direction of the fluid particle’s velocity.7

Equation of state

p = p0 + p′acoustic pressureaccess pressuresound pressure

r = r0 + r'

c2

Equation 1.17 Equation 1.11

p'

r'r0

Linear Eule's equation

acoustic pressure(p')

particle velocity (u)

velocityparticle velocity

density (r0 ,r')

density

∂p'

∂x∂u∂t

– =

Equation 1.14

∂r'

∂t∂u∂x

= – r0

Figure 1.3 Pictorial relation between three variables that govern acoustic wave propagation (p0 and ρ0 expressthe mean pressure and static density, respectively; p′ and ρ ′ denote acoustic pressure and fluctuating density,respectively; c denotes the speed of propagation, and u is the velocity of the fluctuating medium)

6 If we eliminate p′ and u or change to p′ and ρ′, then we can obtain the equation for ρ′ and u, respectively.7 The linearized Euler equation (Equation (1.11)) essentially states that the pressure difference induces time rateof velocity u in the x direction.